15 WEAK TOPOLOGIES 79 4. Let E be a separable real Frechet space and p a continuous seminorm on E. (a) Let (>);> o be a total (12.13) free sequence in E, and let En be the subspace of dimension n -f 1 generated by a0 , ai9 . . . , an . Show by induction on n that for each n there exists a linear form/, on En such that/n is the restriction of fn+1 to £, fo(a0)=p(a0) and fn(x) ^p(x) for all #eEn. (Consider the hyperplane H in En_j given by the equation fn-i(x) = fn-i(a0), and use Problem 3(d) in the plane which is the quotient of En by the hyperplane H0 : fn-i(x) = 0 in Ert «i.) (b) Deduce from (a) that there exists a continuous linear form / on E such that /(flo) = jp(flo) and \f(x)\ ^p(x) for all x e E (Hahn-Banach theorem for a separable Frechet space). In particular, if E is a separable Banach space, then for each x e E we have (the norm on the dual E' being that defined in (5.7.3)). Deduce that if E and F are separable Banach spaces, E' and F' their duals, and u : E -* F a continuous linear mapping, then IMI = \\u\\. (c) Deduce from (b) the geometrical form of the Hahn-Banach theorem : given any nonempty convex open set A in E, and a point x0 £ A, there exists a closed hyperplane with equation g(x) a = 0 such that g(x) > oc for all x e A, and g(xQ) <| a. (Reduce to the case where 0 6 A and XQ is a frontier point of A. We can then assume that A is symmetrical about 0 and therefore is the set of all x such that p(x) < 1, where p is a continuous seminorm on E.) (d) Let A, B be two disjoint closed convex sets in E, one of which (say, A) is compact. Show that there exists a closed hyperplane H, with equation g(x) a = 0, which strictly separates A and B: that is to say, such that g(x) > a for all A: e A and g(x) < oc for all x e B. (e) Deduce from (d) that every closed convex set in E is an intersection of closed half-spaces, and that every closed linear variety in E is an intersection of closed hyperplanes. (f) Deduce that for a vector subspace F not to be dense in E it is necessary and sufficient that there should exist a continuous linear form/ ^ 0 on E such that/Cx) = 0 for all x e F. 5. Let A be a convex set in a real vector space. A point a e A is said to be an extremal point of A if there exists no line-segment containing more than one point, which is contained in A and has a as an interior point: in other words, if the relations b e A, c e A, a = Xb H- (1 A)c, 0 < A < 1 imply b = c ~ a. (a) Let E be a Hilbert space, A a bounded closed convex set in E, 8 > 0 the diameter of A, and a, b two points of A such that \\b a\\ S> JA/15 8. Show that there exists a supporting hyperplane H of A orthogonal to the vector b a and such that the set H n A has diameter gi$. (b) Deduce from (a) that A has at least one extremal point (proof by induction). (c) Show that A is the closed convex hull of the set of its extremal points (Krein- Milman theorem for Hilbert space). (First deduce from (b) that every closed hyper- plane of support of A contains at least one extremal point. Then argue by contradic- tion, assuming that the closed convex hull B of the set of extremal points of A is distinct from A; by using Problem 4(e), show that there would then exist a hyperplane of support of A which did not meet B.) (Cf. Section 13.10, Problem 8.)onvex hull (Section 12.14, Problem 13) of the set of points ±en/n. Show